The theory of compressive sensing (CS) has shown us that under certain conditions, a sparse signal can be recovered from a small number of linear incoherent measurements. An effective class of reconstruction algorithms involve solving a convex optimization program that balances the 1 norm of the solution against a data fidelity term. Tremendous progress has been made in recent years on algorithms for solving these 1 minimization programs. These algorithms, however, are for the most part static: they focus on finding the solution for a fixed set of measurements. In this paper, we will present a suite of dynamic algorithms for solving 1 minimization programs for streaming sets of measurements. We consider cases where the underlying signal changes slightly between measurements, and where new measurements of a fixed signal are sequentially added to the system. We develop algorithms to quickly update the solution of several different types of 1 optimization problems whenever these changes occur, thus avoiding having to solve a new optimization problem from scratch. Our proposed schemes are based on homotopy continuation, which breaks down the solution update in a systematic and efficient way into a small number of linear steps. Each step consists of a low-rank update and a small number of matrix-vector multiplicationsvery much like recursive least squares. Our investigation also includes dynamic updating schemes for 1 decoding problems, where an arbitrary signal is to be recovered from redundant coded measurements which have been corrupted by sparse errors.
Index TermsHomotopy, sparse signal recovery, recursive filtering, compressive sensing, 1 norm minimization, 1 decoding, LASSO, Dantzig selector.